Mirror symmetry implies that the higher genus Gromov-Witten (GW), Pandharipande-Thomas (PT) and
certain Donaldson-Thomas (DT) invariants on the A-model side are encoded in generating functions which are related to analytic
sections of line bundles over the complex moduli space on the B-model side constructed from the periods. This relates the
asymptotic behaviour of the invariants as extracted at the point of maximal unipotent monodromy to the behaviour of these sections
near the closest singularities, typically a conifold. Here the relevant analytic information is encoded in closed and open
modular period integrals over the modular domain of $\Gamma_0(N)$. The latter structures are determined by properties
of the degenerate Calabi-Yau motive in the conifold fibre, which have been proven by the fibering out technique with B\”onisch and
Golyshev. We comment on the implications of the asymptotic behaviour of the symplectic invariants for the estimates of the
microscopic entropy of supersymmetric black holes and black rings as investigated recently with Alexandrov and Pioline.